What is $1+Ax+Ax^2+Ax^3+Ax^4+Ax^5$.... an expansion of ??

Question

$$1 + x+ x^2 + x^3 + x^4+ x^5 + x^6...$$ is the expansion of $\frac{1}{1-x}, $

and

$$1 - x + x^2 - x^3 + x^4 - x^5 + x^6...$$ is the expansion of $\frac{1}{1+x}.$

I am trying to figure out of what are these following expansions?

$$1 + Ax + Ax^2 + Ax^3 + Ax^4+ Ax^5 + Ax^6...$$ is the expansion of ????

and

$$1 - Ax + Ax^2 - Ax^3 + Ax^4 - Ax^5 + Ax^6...$$ is the expansion of ????

Thank You

Answer 1

$$1+Ax+Ax^2+...=1+A(x+x^2+...)=1+A(-1+1+x+x^2+...)$$

$$=1+A(-1+\dfrac1{1-x})=1+\dfrac{Ax}{1-x}$$

Answer 2

Your first sum is $(1-A)+A+Ax+Ax^2+\cdots=1-A+\frac{A}{1-x}$; similarly, the second is $1-A+\frac{A}{1+x}$.

Answer 3

As you well said, $\sum_0^\infty x^n=\frac{1}{1-x}$ but you have to consider thet this is only valid for $|x| \in [0,1[$ that said, we can do the following:

$$1+Ax+Ax^2...=1-A+\sum_{n=0}^\infty Ax^n=1-A+\frac{A}{1-x}=\frac{1+x(A-1)}{1-x}$$

Note that with $A=1$ we have our original series back!

Now, we can do the same with the other series like so:

$$1-Ax+Ax^2-Ax^3...=1-A+A-Ax+Ax^2-Ax^3...=1-A+\frac{A}{1+x}=\frac{1+x(1-A)}{1+x}$$